You do not need the AirPort Utility installed on your computer in order to access a shared drive from either a remote location or on the local network. This utility's primary purpose is for administering the base station.
You will need to both enable File Sharing and configure the AirPort for Port Mapping. It is pretty much described step-by-step in the article you referenced in the beginning of your post.
Ok, so let's go over the basic steps to see if anything was missed.
- Start the AirPort Utility > Select the AEBSn, and then, note the IP address shown.
- Select Manual Setup.
- Verify that Connection Sharing = Share a public IP address is selected on the Internet > Internet Connection tab.
- Select Disks, and then, select File Sharing.
- Verify that both the "Enable file sharing" and "Share disks over WAN" options are enabled.
- Verify that Secure Shared Disks = With a disk password. (Recommended)
- Verify that AirPort Disks Guest Access = Not allowed. (Recommended)
- Select Advanced, and then, select the Port Mapping tab.
- Click the plus sign to add a new port mapping.
- For Service, select the "Personal File Sharing" option. (Note: This option would only allow Mac clients to access the AirPort Disks. If you want both Macs and PCs to connect, you would need to leave this field at its default value and enter the appropriate SMB ports to the mix of ports to be opened.)
- In the Public UDP Port(s) and Public TCP Port(s) boxes, type in a 4-digit port number (e.g., 8888) that you choose. In the Private IP Address box, type the internal IP address of your AEBSn that you wrote down in step 1. In the Private UDP Port(s) and Private TCP Port(s) boxes, type 548. Click Continue.
- In the Description box, type a descriptive name like "AirPort Disk File Sharing," and then, click Done.
- Click on Update.
To connect to the shared AirPort Disk from a remote location using a Mac:
- From the Finder > Go > Connect to Server.
- Enter the DynDNS-provided Domain Name or Pubic (WAN-side) IP address of the AEBSn, followed by a colon and the Public port number that you choose in step 11 of the previous procedure. For example: afp://www.mydyndnsdomain.com:8888 or afp://123.456.789.123:8888
- Click Connect.
- You should be prompted for your user name and password. The user name can be anything you like; the password should be the Disk password for the AEBSn that you created in step 6 previously.
- Click Connect.